{"id":185751,"date":"2010-01-15T00:00:00","date_gmt":"2010-12-28T09:39:20","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/twice-ramanujan-sparsifiers-2\/"},"modified":"2016-08-22T11:27:22","modified_gmt":"2016-08-22T18:27:22","slug":"twice-ramanujan-sparsifiers-2","status":"publish","type":"msr-video","link":"https:\/\/www.microsoft.com\/en-us\/research\/video\/twice-ramanujan-sparsifiers-2\/","title":{"rendered":"Twice-Ramanujan Sparsifiers"},"content":{"rendered":"<div class=\"asset-content\">\n<p>We prove that every graph has a spectral sparsifier with a number of edges linear in its number of vertices. As linear-sized spectral sparsifiers of complete graphs are expanders, our sparsifiers of arbitrary graphs can be viewed as generalizations of expander graphs.<\/p>\n<p>In particular, we prove that for every d > 1 and every undirected, weighted graph G = (V,E,w) on n vertices, there exists a weighted graph H=(V,F,&tilde;w) with at most dn edges such that for every<\/p>\n<p>x &isin; R<sup>V<\/sup>,<br \/>\nx<sup>T<\/sup> L<sub>G<\/sub> x &le; x<sup>T<\/sup> L<sub>H<\/sub> x &le; ( d+1+2&radic;d \/ d+1-2&radic;d ) x<sup>T<\/sup> L<sub>G<\/sub> x<\/p>\n<p>where L<sub>G<\/sub> and L<sub>H<\/sub> are the Laplacian matrices of G and H, respectively. Thus, H approximates G spectrally at least as well as a Ramanujan expander with dn\/2 edges approximates the complete graph.<\/p>\n<p>We give an elementary deterministic polynomial time algorithm for constructing H.<\/p>\n<p>Joint work with Josh Batson and Dan Spielman.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>We prove that every graph has a spectral sparsifier with a number of edges linear in its number of vertices. As linear-sized spectral sparsifiers of complete graphs are expanders, our sparsifiers of arbitrary graphs can be viewed as generalizations of expander graphs. In particular, we prove that for every d > 1 and every undirected, [&hellip;]<\/p>\n","protected":false},"featured_media":280667,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","footnotes":""},"research-area":[13561],"msr-video-type":[],"msr-locale":[268875],"msr-post-option":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-185751","msr-video","type-msr-video","status-publish","has-post-thumbnail","hentry","msr-research-area-algorithms","msr-locale-en_us"],"msr_download_urls":"","msr_external_url":"https:\/\/youtu.be\/I1CSdfMW9kw","msr_secondary_video_url":"","msr_video_file":"","_links":{"self":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/185751"}],"collection":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video"}],"about":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/types\/msr-video"}],"version-history":[{"count":0,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/185751\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media\/280667"}],"wp:attachment":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media?parent=185751"}],"wp:term":[{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=185751"},{"taxonomy":"msr-video-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video-type?post=185751"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=185751"},{"taxonomy":"msr-post-option","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-post-option?post=185751"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=185751"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=185751"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}